Pablo Shmerkin
We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor subsets of the real line, equals either the sum of the dimensions or 1, whichever is smaller.
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In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) a hyperbolic basic set which exhibits $C^2$-robust homoclinic tangency of the largest codimension for any higher dimensional manifold, using blenders. The former implies that an analog of Moreira's theorem on Cantor sets in the real line does not hold in higher dimension. The latter solves a question posed by Barrientos and A.Raibekas.
Integer Cantor Sets: Arithmetic Combinatorial Properties
本文研究整数康托集的算术组合性质，探讨了交集性质、均匀分布、多项式均值遍历定理及度量对称相关性等核心问题。
In this paper we prove that among pairs $K,\,K' \subset \mathbb{C}$ of conformal dynamically defined Cantor sets with sum of Hausdorff dimensions $HD(K)+HD(K')>2$, there is an open and dense subset of such pairs verifying $\text{int}(K-K')\neq \emptyset$. This is motivated by the work \cite{MY}, where Moreira and Yoccoz proved a similar statement for dynamically defined Cantor sets in the real line. Here we adapt their argument to the context of conformal Cantor sets in the complex plane, this requires the introduction of several new concepts and a more detailed analysis in some parts of the a…
Cantor sets in higher dimension I: Criterion for stable intersections
本文在高维空间中研究Cantor集的稳定相交判据，并提出构造显式稳定相交Cantor集的方法，适用于Hausdorff维度极小的复杂情形。
Let $C_\la$ and $C_\ga$ be two affine Cantor sets in $\mathbb{R}$ with similarity dimensions $d_\la$ and $d_\ga$, respectively. We define an analog of the Bandt-Graf condition for self-similar systems and use it to give necessary and sufficient conditions for having $\Ha^{d_\la+d_\ga}(C_\la + C_\ga)>0$ where $C_\la + C_\ga$ denotes the arithmetic sum of the sets. We use this result to analyze the orthogonal projection properties of sets of the form $C_\la \times C_\ga$. We prove that for Lebesgue almost all directions $θ$ for which the projection is not one-to-one, the projection has zero $(d_…
Scale Recurrence Lemma and Dimension Formula for Cantor Sets in the Complex Plane
本文将 Moreira-Yoccoz 尺度递归引理推广到复平面康托集的多维共形情形，并据此在适当假设下证明了乘积康托集在函数作用下 Hausdorff 维数的计算公式。
Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections
本文证明了对于正则Cantor集，上盒维数之和小于d时可被小变形分离，且此约束是最佳的。通过构造具有任意小厚度的正则Cantor集对，展示了C^{1+α}稳定交集的条件。
On faithfulness and DP-transformations generated by arithmetic Cantor series expansions
本文研究了由算术康托系列展开生成的DP变换的信仰度条件及其在单位区间上的维数保持特性。
In the present paper, We introduce a pair of middle Cantor sets namely $(C_α, C_β)$ having stable intersection, while the product of their thickness is smaller than one. Furthermore, the arithmetic difference $C_α- λC_β$ contains at least one interval for each nonzero number $λ$.
Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is irrational, then \[ \dim(C_a+C_b) = \min(\dim(C_a) + \dim(C_b),1), \] where $\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\dim(K)+\dim(K') \le 1$ (``geometric resonance''), then there exists $r<1$ such that all contrac…
Infinite linear patterns in sets of positive density
本文研究正密度集合中无限线性模式的性质，推广了Szemerédi定理和密度有限和定理。
We prove that these Cantor sets are made up of transcendental numbers, apart from their endpoints $0$ and $1$, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers $1$, a characteristic Sturmian number, and an arbitrary Sturmian number with the same slope.
We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeomorphic. Hairy Cantor sets appear in the study of the dynamics of holomorphic maps with infinitely many renormalisation structures. They are employed to link the fundamental concepts of polynomial-like renormalisation by Douady-Hubbard with the arithmetic conditions obtained by H…
We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: Let m,n be integers which are not powers of the same integer, and let X,Y be closed subsets of the unit interval which are invariant, respectively, under times-m mod 1 and times-n mod 1. Then, for any non-zero t: dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures, and gives a…
This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…
Let $C_1$ and $C_2$ be two Cantor sets with convex hull $[0,1]$. Newhouse proved if $τ(C_1)\cdot τ(C_2)\geq 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $τ(C_i), 1\leq i\leq 2$ denotes the thickness of $C_i$. In this paper, we generalize this thickness theorem as follows. Let $K_i\subset \mathbb{R}, i=1,\cdots, d$, be some Cantor sets (perfect and nowhere dense) with convex hull $[0,1]$. Suppose $f(x_1,\cdots, x_{d-1},z)\in \mathcal{C}^1$ is a continuous function defined on $\mathbb{R}^d$. Denote the continuous image of $f$ by $$f(K_1,\cdots, K_d)=\{f(x_1, \cdots x_{d-1},z):x_i\…
From Dimension Drop to Aperiodic Order
本文对双几何级数可达集族给出完整拓扑分类，其二元分类诱导出格点上的非周期双色铺砌，并揭示维数下降现象，证明该铺砌是四个算术定义状态上替换铺砌的因子。
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